Coalgebra, Automata, and Document Synchronization Kazuhiro Inaba Reading “Merging Hierarchically-Structured Documents in Workflow Systems” [E. Badouel and M. T. Tchendji, CMCS’08] IPLAS 2009/06/09

((All pictures are cited from the authors presentation slide: http://old-www.cwi.nl/projects/cmcs08/slides/index.html thanks.))

≪Problem≫ Partial views of a (big) document Concurrently updated How to merge them? Anyhow, the problem the authors want to address is this. There is a big document tree, and many people are working on its partial views concurrently. The problem is, how to merge them consitently?

Approach of this paper Use coalgebra (= tree automaton)! Solution is, to use a certain kind of coalgebra, namely, tree automaton,

Represented by an automaton doc u2 u1 view2 view1 Represented by an automaton Set of all doc’s s.t. view2(doc’) = u2 Set of all doc’s s.t. view1(doc’) = u1 Here is the one picture summary of the work. We have an original document ‘doc’, two view functions view1 and view2, and viewed document u1 and u2. They at updated independently To merge them, in the coalgebraic approach, we first compute the set of inverse images. They can be infinite inter- section !!

Basic Notions Document View Grammar Projection Expansion Now let us go into the details. First I introduce some notions, in other words, I’ll exapling how the documents or views are modeled in this work.

(Untyped) Document Let S be a finite alphabet E.g. S = {A,B,C} A “document” (over S) is an unranked tree over S First, a document they consider is a tree, an unranked tree. Unranked means that the nodes with same label may have different number of children

(Untyped) View A “view” is a subset of S The “projection associated with a view” is the function (of type: tree  forest) that erases all symbols not in the view View on a document is defined very simply.

(Untyped) Expansion The “expansion associated with a view” is the function (of type: forest  2tree) which is the inverse of the projection Note: the output set is regular!  they’re represented as a tree automaton How precisely? Wait a moment…

Grammar In the paper, the authors consider only “typed” documents. Typed = Conformance to a grammar A “Grammar” over S is a triple (S , A, P): A ∈ S axiom (initial symbol) P ⊆ S ×S * set of productions

(Typed) Document A grammar G = (S , A, P) corresponds to a set of trees called “derivation trees”, defined for each X∈S as follows: Der(G, X) = {X(t1,…, tn) | ∃XX1…Xn ∈ P: ∀i: ti∈Der(G,Xi)} The members of Der(G, A) are the document confirming the grammar G From now on, we deal with such docs only

Example of a Grammar & a Doc.

(Typed) View Same as before “expansion” takes two more parameters A “view” ⊆ S, “projection” is an erasure “expansion” takes two more parameters G = (S, A, P) : Grammar X ∈ S : Axiom expansion(V, ts, G, X) returns the set of trees in Der(G,X) whose projection with V are equal to ts

Example of an Expansion

Top-Down Nodeterministic Tree Automata Tree Automaton A is a tuple <S, Q, δ>: S : node labels Q : set of states δ : Q  2S ×Q* : transition relation Grammars are straightforwardly converted to tree automata: G = (S , A, P) ~~> A = (S, Q, δ) where Q = S δ( q ) = {(q, q1 … qn) | qq1…qn ∈ P}

Example A = ({A,B}, {q1,q2}, δ) q1 q2 q2 q1 q2 q2 δ(q1) = { (A, q1q2), (A, q2q2) } δ(q2) = { (B, ), (B, q1) } q1 This tree is in the set of trees repr’d by (A, q1)! A A q2 q2 B B B A q1 A B B This tree is not! (no possible assignment) q2 q2 B B

Representing Expansions by TA Recall: expansion(V, ts, G, X) returns the set of trees in Der(G,X) whose projection by V are equal to ts. V ⊆ S G = (S , P)

Input: V, G=(S, P), ts Output: the corresponding automaton Corresponding automaton is A = (S, Q, δ): Q = S ×T T is the list of subtrees of ts δ( <s, t> ) = Φ if s∈V and t≠[s(…)] { (s, <s1, t1><s2, t2>…<sn, tn>) | ss1…sn ∈ P, t1’ … tn’ = t’ } if s ∈V and t=[s(t’)] or s ∉ V and t=t’

Proposition (Correctness) Member of expansion(V, ts, G, X) iff Accepted by the automaton A from the state <X, ts>

Example S = {A,B,C} G = (S, A, P) where P is: V = {A, B} ts =

δ( <A, A(A,B(A,A))> ) = { (A, <C, A,B(A,A))><B,ε>), (A, <C, A><B, B(A,A))>), (A, <C, ε><B, A,B(A,A)) >) } δ( <C, A,B(A,A))> ) = { (C, <A, A,B(A,A)><C, ε>), (C, <A, A><C, B(A,A)>), (C, <A, ε><C, A,B(A,A)>), (C, <C, A,B(A,A)><C, ε>), (C, <C, A><C, B(A,A)>), (C, <C, ε><C, A,B(A,A)>) } …

Now, On Tree Automata… We can compute Emptiness of the represented set Coinductively Intersection between two sets By Product Construction (Q, δ) ∩ (P, γ) = (Q×P, β) where β( (q, p) ) = { (σ, (q1,p1)…(qn,pn)) | (σ,q1…qn)∈δ(q) and (σ,p1…pn)∈γ(p) }

Algorithm: “Coherence” check Given Grammar G = (S, A, P) View V1 and Forest ts1 View V2 and Forest ts2 Are these two views coherent? (i.e., can we “merge” them into a single document that generates the two views simultaneously?) expansion(V1,ts1,G,A) ∩ expansion(V2,ts2,G,A) ≠ Φ?

Algorithm: “Synchronization” Given Grammar G = (S, A, P) View V1 and Forest ts1 View V2 and Forest ts2 How can we get the merged document?  If is a singleton set, that’s it! Otherwise, ambiguous  error expansion(V1,ts1,G,A) ∩ expansion(V2,ts2,G,A)

Singleton check (Not in the paper…) Step 1 (Cleaning): A ~~> Acl Eliminate all “failure” states and transitions Step 2 (Thinning): Acl ~~> Acl,th Determinize the automaton by dropping nondeterministic rules Step 3 (Equivalence Check) Acl =? Acl,th Check Bisimilarity

Example 1 ADDRESSBOOK  @ @  ε | PERSON @ PERSON  NAME ADDRESS TEL V1 = {NAME} V2 = {TEL}

Example of a document: ADDRESSBOOK[@[ PERSON[ NAME[…] ADDRESS[…] TEL[…] ]@[ ]@[]]]] expansion(V1, NAME[…]NAME[…]) is consistent with expansion(V2, TEL[…]TEL[…]) but not with expansion(V2, TEL[…]TEL[…]TEL[…])

Example 2 ADDRESSBOOK  @ @  ε | PERSON @ PERSON  NAME ADDRESS TEL # V1 = {NAME} V2 = {TEL}

Example of a document: ADDRESSBOOK[@[ PERSON[ NAME[…] ADDRESS[…] TEL[…] #[] ]@[ NAME[…] ADDRESS[…] TEL[…] #[TEL[…] #[]] ]@[]]]] expansion(V1, NAME[…]NAME[…]) is consistent with expansion(V2, TEL[…]TEL[…]) and also with expansion(V2, TEL[…]TEL[…]TEL[…])

Ambiguity Example of a document: ADDRESSBOOK[@[ PERSON[ NAME[…] ADDRESS[…] #[TEL[…] #[]] ]@[ NAME[…] ADDRESS[…] #[TEL[…] #[TEL[…] #[]] ]@[]]]] Ambiguity in the synchronization of expansion(V1, NAME[…]NAME[…]) with expansion(V2, TEL[…]TEL[…]TEL[…])

How to resolve ambiguity Be more careful in choosing views Set of views that “covers” the whole structure Such as V1 = {NAME} V2 = {PERSON, TEL} or V1 = {PERSON, NAME} V2 = {TEL, #} etc.

“Static” Singleton Checking? How can the merging system support users to choose appropriate views? For example, Given a set of views, can we check whether they cause ambiguity or not?  Undecidable Problem (reduces to the ambiguity of CFG) (No clear solution is given in the paper…)

Summary Synchronization of multiple (edited) views can be computed by using tree automaton Compute inverse-image of the view Compute intersection, emptiness, & singularity Further topic (in the paper, but not in this presentation) “To-be-written” node For each symbol X in the grammar, add X and Xε Synchronizer allows X in u2 to occur at X position in u1, etc…

Appendix: Automata as Coalgebra J.J.M.M. Rutten, “Automata and Coinduction (An Exercise in Coalgebra)”, CONCUR 1998 The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.